The nonlinear membrane model: a Young measure and varifold formulation

¹ Centre Universitaire de Médéa, Institut des Sciences de l'Ingénieur, CC 151, Quartier Ain-D'Heb, Médéa (26000), Algeria.
² Laboratoire de Mécanique et de Génie Civil, UMR-CNRS 5508, Université Montpellier II, Case courier 048, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France.
³ ACSIOM et EMIAN, UMR-CNRS 5149, Université Montpellier 2 et CUFR de Nîmes, Case courier 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France.

Received: 11 February 2004

Abstract

We establish two new formulations of the membrane problem by working in the space of -Young measures and -varifolds. The energy functional related to these formulations is obtained as a limit of the 3d formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. The interest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences related to the classical formulation. The second formulation moreover accounts for concentration effects.